A Ferris wheel of radius 100 feet is rotating at a constant angular speed ? rad/

Question

A Ferris wheel of radius 100 feet is rotating at a constant angular speed ? rad/sec counterclockwise. Using a stopwatch, the rider finds it takes 6 seconds to go from the lowest point on the ride to a point Q, which is level with the top of a 44 ft pole. Assume the lowest point of the ride is 3 feet above ground level.

Let Q(t)=(x(t),y(t)) be the coordinates of the rider at time t seconds; i.e., the parametric equations. Assuming the rider begins at the lowest point on the wheel, then the parametric equations will have the form: x(t)=rcos(?t-?/2) and y(t)=rsin(?t -?/2), where r,? can be determined from the information given. Provide answers below accurate to 3 decimal places. (Note: We have imposed a coordinate system so that the center of the ferris wheel is the origin. There are other ways to impose coordinates, leading to different parametric equations.)

Explanation / Answer

Find the change in angle for 6 seconds.

cos(??) = (100 + 3 - 44 ft) / (100 ft)

?? = 0.9397375 rad

Calculate ?.

? = ??/?t

? = (0.9397375 rad) / (6 s)

? = 0.156623 rad/s

Parametric equation for y:

y(t) = 3 ft + (100 ft)sin(0.156623t - ?/2)

Find first 2 times for y = 80 ft.

80 = 103 + 100 sin(0.156623t - ?/2)

-0.23 = sin(0.156623t - ?/2)

-0.232078 + 2?n = 0.156623t - ?/2 or (? + 0.232078) + 2?n = 0.156623t - ?/2

First time:

-0.232078 = 0.156623t - ?/2

1.338718 = 0.156623t

t = 8.547 s

Second time:

? + 0.232078 = 0.156623t - ?/2

4.944467 = 0.156623t

t = 31.569 s

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